Notes: You are on the Moon, observing Earth, at a distance of about 384,000 kilometers. Using the small-angle formula you conclude that the actual...

Wednesday, May 6, 2009

You are on the Moon, observing Earth, at a distance of about 384,000 kilometers. Using the small-angle formula you conclude that the actual...

Denote the distance between the Moon and Earth as $\displaystyle{D}$ and the diameter of Earth as $\displaystyle{d}.$ We have an isosceles triangle with the legs of the length $\displaystyle{D}$ and the base of the length $\displaystyle{d}.$ Denote the angle between the legs as $\displaystyle\alpha,$ it is a small angle and we have to find it.

Then is is clear that $\displaystyle{\tan{{\left(\frac{\alpha}{{2}}\right)}}}=\frac{{\frac{{d}}{{2}}}}{{D}}=\frac{{d}}{{{2}{D}}}.$

It is true that for small angles $\displaystyle\alpha$ $\displaystyle{\tan{{\left(\alpha\right)}}}\approx{\sin{{\left(\alpha\right)}}}\approx\alpha.$ Therefore $\displaystyle{\tan{{\left(\frac{\alpha}{{2}}\right)}}}=\frac{{d}}{{{2}{D}}}$ implies $\displaystyle\frac{\alpha}{{2}}\approx\frac{{d}}{{{2}{D}}},$ or $\displaystyle\alpha\approx\frac{{d}}{{D}}.$ In numbers it is equal to $\displaystyle\frac{{{12},{756}}}{{{384},{000}}}\approx{0.033}$ radians. In degrees it is about $\displaystyle{1.9}.$